\begin{align} \int \frac{x^3 + 3x + 4}{\sqrt{x}} . dx\end{align}
\begin{align} =\int \left(x^\frac{5}{2} + 3x^\frac{1}{2} + 4x^\frac{1}{2}\right) . dx\end{align}
\begin{align} =\frac{\left(x^{\displaystyle\frac72}\right)}{\displaystyle\frac72}+ \frac{3\left(x^{\displaystyle\frac32}\right)}{\displaystyle\frac32} + \frac{4\left(x^{\displaystyle\frac12}\right)}{\displaystyle\frac12} + C\end{align}
\begin{align} =\frac27\left(x^\frac72\right)+ 2\left(x^\frac32\right) + 8\left(x^\frac12\right) + C\end{align}
\begin{align} =\frac27\left(x^\frac72\right)+ 2\left(x^\frac32\right) + 8\left(\sqrt x\right) + C\end{align}
In each of the following cases, state whether the function is one-one, onto or bijective. Justify your answer.
(i) f : R → R defined by f(x) = 3 – 4x
(ii) f : R → R defined by f(x) = 1 + x2
Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.
Prove that the Greatest Integer Function f : R → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.
Show that the Modulus Function f : R → R, given by f(x) = |x|, is neither oneone nor onto, where | x | is x, if x is positive or 0 and |x| is – x, if x is negative.
Consider f : R+ → [4, ∞) given by f(x) = x2 + 4. Show that f is invertible with the inverse f–1 of f given by , where R+ is the set of all non-negative real numbers.
Let f: X → Y be an invertible function. Show that the inverse of f –1 is f, i.e., (f–1)–1 = f.
A balloon, which always remains spherical, has a variable diameter
\begin{align} \frac{3}{2}(2x+1)\end{align}
Find the rate of change of its volume with respect to x.
Show that the Signum Function f : R → R, given by
is neither one-one nor onto.